Lecture 1 - First Order Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


 
 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: First Order Differential Equations 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
Page 2


 
 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: First Order Differential Equations 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
Table of Contents 
Chapter: First Order Differential Equations 
1. Learning Outcomes 
2. Introduction 
3. Basic terminology 
4. Integral as General, Particular and singular solutions 
5. Separable equations 
6. Exercise 
7. Differential Equations and Mathematical Models 
7.1 Application in coordinate geometry 
7.2 Applications of Differential Equation in science and Engineering 
7.3  Torricelli’s Law 
7.4  Newton’s Law of Cooling 
7.5  Growth and Decay- Population growth, radio -active decay, drug 
assimilation, compound interest . 
8. Exercise 
Summary 
References 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
Page 3


 
 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: First Order Differential Equations 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
Table of Contents 
Chapter: First Order Differential Equations 
1. Learning Outcomes 
2. Introduction 
3. Basic terminology 
4. Integral as General, Particular and singular solutions 
5. Separable equations 
6. Exercise 
7. Differential Equations and Mathematical Models 
7.1 Application in coordinate geometry 
7.2 Applications of Differential Equation in science and Engineering 
7.3  Torricelli’s Law 
7.4  Newton’s Law of Cooling 
7.5  Growth and Decay- Population growth, radio -active decay, drug 
assimilation, compound interest . 
8. Exercise 
Summary 
References 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of an ordinary differential equation, and partial differential 
equation.  
• How to find the order and degree of a differential equation. 
• Solution of a differential equation. 
• Types of solutions of a differential equation-General, particular and 
singular solution. 
• Separable Equations. 
• Use of differential equations in coordinate geometry. 
• Mathematical modelling of Newton’s Law of Cooling and Torricelli’s 
Law and its applications. 
• Growth and decay problems - population growth, radio -active 
decay, drug assimilation and compound interest problems. 
• Concept of velocity and acceleration. 
 
2. Introduction 
Differential equations finds its application in a variety of real world 
problems such as growth and decay problems. Newton’s law of cooling 
can be used to determine the time of death of a person. Torricelli’s law 
can be used to determine the time when the tank gets drained off 
completely and many other problems in science and engineering can be 
solved by using differential equations. In this chapter, we will first discuss 
the concept of differential equations and the method of solving a first 
order differential equation. In the next section, we will discuss various 
applications of differential equations. 
3. Basic Terminology 
Variable: Variable is that quantity which takes on different quantitative 
values. Example: memory test scores, height of individuals, yield of rice 
etc. 
Dependent Variable: A variable that depends on the other variable is 
called a dependent variable. For instance, if the demand of gold depends 
on its price, then demand of gold is a dependent variable. 
Independent Variable: Variables which takes on values independently 
are called independent variables. In the above example, price is an 
independent variable.  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
Page 4


 
 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: First Order Differential Equations 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
Table of Contents 
Chapter: First Order Differential Equations 
1. Learning Outcomes 
2. Introduction 
3. Basic terminology 
4. Integral as General, Particular and singular solutions 
5. Separable equations 
6. Exercise 
7. Differential Equations and Mathematical Models 
7.1 Application in coordinate geometry 
7.2 Applications of Differential Equation in science and Engineering 
7.3  Torricelli’s Law 
7.4  Newton’s Law of Cooling 
7.5  Growth and Decay- Population growth, radio -active decay, drug 
assimilation, compound interest . 
8. Exercise 
Summary 
References 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of an ordinary differential equation, and partial differential 
equation.  
• How to find the order and degree of a differential equation. 
• Solution of a differential equation. 
• Types of solutions of a differential equation-General, particular and 
singular solution. 
• Separable Equations. 
• Use of differential equations in coordinate geometry. 
• Mathematical modelling of Newton’s Law of Cooling and Torricelli’s 
Law and its applications. 
• Growth and decay problems - population growth, radio -active 
decay, drug assimilation and compound interest problems. 
• Concept of velocity and acceleration. 
 
2. Introduction 
Differential equations finds its application in a variety of real world 
problems such as growth and decay problems. Newton’s law of cooling 
can be used to determine the time of death of a person. Torricelli’s law 
can be used to determine the time when the tank gets drained off 
completely and many other problems in science and engineering can be 
solved by using differential equations. In this chapter, we will first discuss 
the concept of differential equations and the method of solving a first 
order differential equation. In the next section, we will discuss various 
applications of differential equations. 
3. Basic Terminology 
Variable: Variable is that quantity which takes on different quantitative 
values. Example: memory test scores, height of individuals, yield of rice 
etc. 
Dependent Variable: A variable that depends on the other variable is 
called a dependent variable. For instance, if the demand of gold depends 
on its price, then demand of gold is a dependent variable. 
Independent Variable: Variables which takes on values independently 
are called independent variables. In the above example, price is an 
independent variable.  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
Derivative: Let y = f(x) be a function. Then the derivative 
???????? 
???????? = ????' ( ???? ) of 
the function f is the rate at which the function y = f(x) is changing with 
respect to the independent variable. 
Differential Equation: An equation which relates an independent 
variable, dependent variable and one or more of its derivatives with 
respect to independent variable is called a differential equation. 
Example1: 
???????? ???????? = 2 ???? is a differential equation which involves an 
independent variable x, dependent variable y, first derivative of y with 
respect to x. This equation involves the unknown function y of the 
independent variable x and first derivative 
???????? ???????? of y w.r.t. x 
Example2: 
2
2
3 20
d y dy
y
dx dx
- +=  is a differential equation which consists of 
an unknown function y of the independent variable x and the first two 
derivatives 
dy
dx
 and 
2
2
dy
dx
 of y w.r.t. x. 
Value Addition: Ordinary and Partial Differential Equations 
Ordinary differential equation: A differential equation in which the 
dependent variable (unknown function) depends only on a single 
independent variable is called an ordinary differential equation. 
Partial Differential equation: A differential equation in which the 
dependent variable is a function of two or more independent variables is 
called a partial differential equation. 
 
Order of a differential equation: The order of a differential equation is 
defined as the order of the highest order derivative appearing in the 
differential equation. The order of a differential equation is a positive 
integer. 
Example3: In the differential equation 
2
4
3
3
30
d y dy
dx dx
??
? ?
+ +=
??? ?
? ?
??
, the order of 
the highest order derivative is 3, so it is a differential equation of order 3. 
Value Addition: First order differential equation 
A differential equation of the form (, )
dy
f xy
dx
= is called a differential 
equation of first order. If initial condition 
00
() yx y =  is also specified , then 
it is called an initial value problem. 
 
 
Value Addition: n
th
 order differential equation 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
Page 5


 
 
 
 
 
Discipline Course – I 
Semester :II 
Paper: Differential Equations - I 
Lesson: First Order Differential Equations 
Lesson Developer: Dr. Kavita Gupta 
College: Ramjas College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
Table of Contents 
Chapter: First Order Differential Equations 
1. Learning Outcomes 
2. Introduction 
3. Basic terminology 
4. Integral as General, Particular and singular solutions 
5. Separable equations 
6. Exercise 
7. Differential Equations and Mathematical Models 
7.1 Application in coordinate geometry 
7.2 Applications of Differential Equation in science and Engineering 
7.3  Torricelli’s Law 
7.4  Newton’s Law of Cooling 
7.5  Growth and Decay- Population growth, radio -active decay, drug 
assimilation, compound interest . 
8. Exercise 
Summary 
References 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
 
1. Learning Outcomes: 
After studying this chapter, you will be able to understand: 
• Meaning of an ordinary differential equation, and partial differential 
equation.  
• How to find the order and degree of a differential equation. 
• Solution of a differential equation. 
• Types of solutions of a differential equation-General, particular and 
singular solution. 
• Separable Equations. 
• Use of differential equations in coordinate geometry. 
• Mathematical modelling of Newton’s Law of Cooling and Torricelli’s 
Law and its applications. 
• Growth and decay problems - population growth, radio -active 
decay, drug assimilation and compound interest problems. 
• Concept of velocity and acceleration. 
 
2. Introduction 
Differential equations finds its application in a variety of real world 
problems such as growth and decay problems. Newton’s law of cooling 
can be used to determine the time of death of a person. Torricelli’s law 
can be used to determine the time when the tank gets drained off 
completely and many other problems in science and engineering can be 
solved by using differential equations. In this chapter, we will first discuss 
the concept of differential equations and the method of solving a first 
order differential equation. In the next section, we will discuss various 
applications of differential equations. 
3. Basic Terminology 
Variable: Variable is that quantity which takes on different quantitative 
values. Example: memory test scores, height of individuals, yield of rice 
etc. 
Dependent Variable: A variable that depends on the other variable is 
called a dependent variable. For instance, if the demand of gold depends 
on its price, then demand of gold is a dependent variable. 
Independent Variable: Variables which takes on values independently 
are called independent variables. In the above example, price is an 
independent variable.  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
Derivative: Let y = f(x) be a function. Then the derivative 
???????? 
???????? = ????' ( ???? ) of 
the function f is the rate at which the function y = f(x) is changing with 
respect to the independent variable. 
Differential Equation: An equation which relates an independent 
variable, dependent variable and one or more of its derivatives with 
respect to independent variable is called a differential equation. 
Example1: 
???????? ???????? = 2 ???? is a differential equation which involves an 
independent variable x, dependent variable y, first derivative of y with 
respect to x. This equation involves the unknown function y of the 
independent variable x and first derivative 
???????? ???????? of y w.r.t. x 
Example2: 
2
2
3 20
d y dy
y
dx dx
- +=  is a differential equation which consists of 
an unknown function y of the independent variable x and the first two 
derivatives 
dy
dx
 and 
2
2
dy
dx
 of y w.r.t. x. 
Value Addition: Ordinary and Partial Differential Equations 
Ordinary differential equation: A differential equation in which the 
dependent variable (unknown function) depends only on a single 
independent variable is called an ordinary differential equation. 
Partial Differential equation: A differential equation in which the 
dependent variable is a function of two or more independent variables is 
called a partial differential equation. 
 
Order of a differential equation: The order of a differential equation is 
defined as the order of the highest order derivative appearing in the 
differential equation. The order of a differential equation is a positive 
integer. 
Example3: In the differential equation 
2
4
3
3
30
d y dy
dx dx
??
? ?
+ +=
??? ?
? ?
??
, the order of 
the highest order derivative is 3, so it is a differential equation of order 3. 
Value Addition: First order differential equation 
A differential equation of the form (, )
dy
f xy
dx
= is called a differential 
equation of first order. If initial condition 
00
() yx y =  is also specified , then 
it is called an initial value problem. 
 
 
Value Addition: n
th
 order differential equation 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
A differential equation of the form 
( ) ( ) 2 (1)
( , , , ,........ ) 0
n
F xy y y y = where F is a 
real valued function of n+2 variables, x is an independent variable and y 
is a dependent variable is called an n
th
 order differential equation. 
 
Degree of a differential equation: The exponent of the highest order 
derivative appearing in the differential equation, when all derivatives are 
made free from radicals and fractions, is called degree of the differential 
equation. In other words, it is the power of the highest order derivative 
occurring in a differential equation when it is written as a polynomial in 
derivatives. 
Example 4: In the differential equation
24
32
32
6 40
dy d y
y
dx dx
? ?? ?
- -=
? ?? ?
? ?? ?
, the 
highest order derivative is 
3
3
dy
dx
 and its exponent or power is 2. So, it is a 
differential equation of order 3 and degree 2. 
Example 5: Consider the differential equation 
1/3
2
2
2
1
dy d y
c
dx dx
??
? ?
+ =
?? ? ?
? ?
??
. To 
express the differential equation as a polynomial in derivatives, we 
proceed as follows: 
Squaring both sides, we get 
 
2/3
2
2
2
1
dy d y
c
dx dx
??
? ?
+ =
?? ? ?
? ?
??
 
Cubing both sides , we get 
 
3
2
2
2
2
1
dy d y
c
dx dx
??
??
? ?
+=
??
?? ? ?
? ?
?? ??
??
 
 
2
624
2
2
2
1 33
dy dy dy d y
c
dx dx dx dx
? ?
? ? ? ? ? ?
? + + + =
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ?
 
 
2
642
2
2
2
3 3 10
d y dy dy dy
c
dx dx dx dx
? ?
? ? ? ? ? ?
? - ---=
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ?
 
Now, the highest order derivative appearing in the polynomial form of the 
given differential equation is 
2
2
dy
dx
. Its exponent is 2. Therefore, degree of 
the given differential equation is 2. Infact, its order is also 2. 
Institute of Lifelong Learning, University of Delhi                                                 pg. 5